nLab bifibrant object



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks



An object in a model category is bifibrant if it is both fibrant as well as cofibrant.

To some extent a model category-structure on a homotopical category/relative category may be understood as a device for finding/forming the full subcategory of bifibrant objects and with it a convenient presentation of the localization at the given class of weak equivalences:

This makes bifibrant objects a convenient notion for abstract reasoning about (simplicial) localizations.

On the other hand, in most model categories the bifibrant objects are not the ones that are conveniently handled in practice (often they can be produced only by abstract fibrant+cofibrant resolution-machines which tend to produce unwieldy results). But the tools provided by model category-structure also allow to reason about bifibrant objects without necessarily constructing them. For example, a key lemma says that for computing homotopy classes of maps between bifibrant objects it is actually sufficient to use an equivalent cofibrant object for the domain and an equivalent fibrant object for the codomain.


See at model category.

Last revised on February 23, 2024 at 19:35:44. See the history of this page for a list of all contributions to it.